| L(s) = 1 | + 72·3-s + 1.76e3·9-s + 1.64e3·11-s − 6.26e4·25-s + 9.72e3·27-s + 1.18e5·33-s − 4.17e5·49-s + 8.36e5·59-s − 1.96e6·73-s − 4.50e6·75-s − 1.28e5·81-s − 5.87e5·83-s − 1.47e6·97-s + 2.90e6·99-s − 7.65e5·107-s − 1.21e7·121-s + 127-s + 131-s + 137-s + 139-s − 3.00e7·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.38e7·169-s + ⋯ |
| L(s) = 1 | + 8/3·3-s + 2.41·9-s + 1.23·11-s − 4.00·25-s + 0.493·27-s + 3.30·33-s − 3.55·49-s + 4.07·59-s − 5.05·73-s − 10.6·75-s − 0.242·81-s − 1.02·83-s − 1.61·97-s + 2.99·99-s − 0.624·107-s − 6.86·121-s − 9.47·147-s + 2.87·169-s + 10.8·177-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(27.68389387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.68389387\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - 4 p^{2} T + 118 p^{2} T^{2} - 4 p^{8} T^{3} + p^{12} T^{4} )^{2} \) |
| good | 5 | \( ( 1 + 1252 p^{2} T^{2} + 5863326 p^{3} T^{4} + 1252 p^{14} T^{6} + p^{24} T^{8} )^{2} \) |
| 7 | \( ( 1 + 208996 T^{2} + 30837454086 T^{4} + 208996 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 11 | \( ( 1 - 412 T + 314898 p T^{2} - 412 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 13 | \( ( 1 - 6934756 T^{2} + 21911820809766 T^{4} - 6934756 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 17 | \( ( 1 - 64896772 T^{2} + 2198931719582598 T^{4} - 64896772 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 19 | \( ( 1 - 177090628 T^{2} + 12242150590475238 T^{4} - 177090628 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 23 | \( ( 1 - 1858756 T^{2} + 43746394215781446 T^{4} - 1858756 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 29 | \( ( 1 + 384071044 T^{2} + 262631660267478246 T^{4} + 384071044 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 31 | \( ( 1 + 2960240164 T^{2} + 3713060857169843526 T^{4} + 2960240164 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 37 | \( ( 1 - 3235129636 T^{2} + 7977254229970771686 T^{4} - 3235129636 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 41 | \( ( 1 - 13543268548 T^{2} + 89790600389496386118 T^{4} - 13543268548 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 43 | \( ( 1 - 6754505476 T^{2} + 39481734937898737446 T^{4} - 6754505476 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 47 | \( ( 1 - 13978081156 T^{2} + \)\(16\!\cdots\!86\)\( T^{4} - 13978081156 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 53 | \( ( 1 + 52183699396 T^{2} + \)\(13\!\cdots\!66\)\( T^{4} + 52183699396 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 59 | \( ( 1 - 209156 T + 54505500486 T^{2} - 209156 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 61 | \( ( 1 - 97087344484 T^{2} + \)\(75\!\cdots\!26\)\( T^{4} - 97087344484 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 67 | \( ( 1 - 172396628932 T^{2} + \)\(20\!\cdots\!98\)\( T^{4} - 172396628932 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 71 | \( ( 1 - 254155821124 T^{2} + \)\(48\!\cdots\!46\)\( T^{4} - 254155821124 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 73 | \( ( 1 + 491236 T + 352157486502 T^{2} + 491236 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 79 | \( ( 1 - 65015134556 T^{2} + \)\(11\!\cdots\!46\)\( T^{4} - 65015134556 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 83 | \( ( 1 + 146756 T + 654996035622 T^{2} + 146756 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 89 | \( ( 1 - 329765563588 T^{2} + \)\(84\!\cdots\!58\)\( T^{4} - 329765563588 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 97 | \( ( 1 + 369308 T - 106588984506 T^{2} + 369308 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.75062048542120879930852186103, −3.69862030945152495086257884631, −3.69240342868416630644234511848, −3.66914437623415541740821447421, −3.58316426539692895773471181638, −3.10806660109886452526604885942, −2.84096845072587382258574324287, −2.77936684170640143876111215163, −2.75289211120022500044542645836, −2.72250895567373298153076109425, −2.50881242992071852875557088943, −2.48267700728745436263743968629, −2.28220394436603160555556271840, −1.74830463991002463042862358362, −1.67007992751173446967369257325, −1.66344512127844876337068324884, −1.61342733323453819143261748384, −1.61338158101319699381294094841, −1.32574043181059188871173583171, −1.13466844677818076563660160991, −0.77509946265071800688738944716, −0.46828845265397081747307540755, −0.40199812721119533021690682582, −0.28693151233784280592177870561, −0.26179773600980024505256136564,
0.26179773600980024505256136564, 0.28693151233784280592177870561, 0.40199812721119533021690682582, 0.46828845265397081747307540755, 0.77509946265071800688738944716, 1.13466844677818076563660160991, 1.32574043181059188871173583171, 1.61338158101319699381294094841, 1.61342733323453819143261748384, 1.66344512127844876337068324884, 1.67007992751173446967369257325, 1.74830463991002463042862358362, 2.28220394436603160555556271840, 2.48267700728745436263743968629, 2.50881242992071852875557088943, 2.72250895567373298153076109425, 2.75289211120022500044542645836, 2.77936684170640143876111215163, 2.84096845072587382258574324287, 3.10806660109886452526604885942, 3.58316426539692895773471181638, 3.66914437623415541740821447421, 3.69240342868416630644234511848, 3.69862030945152495086257884631, 3.75062048542120879930852186103
Plot not available for L-functions of degree greater than 10.
